INTERMEDIARIES IN BREDON (CO)HOMOLOGY AND CLASSIFYING SPACES

FOTINI DEMBEGIOTI, NANSEN PETROSYAN, AND OLYMPIA TALELLI

Abstract. For certain contractible G-CW-complexes and 𝔉 a family of subgroups of G, we construct a spectral sequence converging to the 𝔉-Bredon cohomology of G with E_{1}-terms given by the 𝔉-Bredon cohomology of the stabilizer subgroups. As applications, we obtain several corollaries concerning the cohomological and geometric dimensions of the classifying space E_{𝔉}G. We also introduce, for any subgroup closed class of groups 𝔉, a hierarchically defined class of groups and show that if a group G is in this class, then G has finite 𝔉 ∩ G-Bredon (co)homological dimension if and only if G has jump 𝔉 ∩ G-Bredon (co)homology.

1. Introduction

In [3], Bredon introduced a cohomology theory for equivariant CW-complexes. One of the motivations was to develop an obstruction theory for equivariant extension of maps. This cohomology was further studied by many authors with applications to proper group actions and classifying spaces for families of subgroups (e.g. see Lück [18]).

For a collection of subgroups 𝔉 of a group G that is closed under conjugation and finite intersections, one can consider the homotopy category of G-CW-complexes with stabilizers in 𝔉. A terminal object in this category is called a model for the classifying space E_{𝔉}G, and one can show that these models always exist under the given assumptions on 𝔉 (e.g. see [18]). The augmented cellular chain complex of E_{𝔉}G yields a projective resolution which can be used to compute 𝔉-Bredon cohomology of G. In relation to Baum-Connes and Farell-Jones Isomorphism conjectures (see [2], [22], [6]), there has been extensive research on questions concerning finiteness properties of E_{𝔉}G, in particular when 𝔉 is the family of finite subgroups or the family of virtually cyclic subgroups (e.g. see [15],[16],[19],[23]).

The purpose of this paper is twofold. First, given a G-CW-complex X such that for each F ∈ 𝔉 the subcomplex X^{F} is contractible, we construct a spectral sequence converging to the 𝔉-Bredon cohomology of G. The E_{1}-terms of the spectral sequence are given in terms

Date: June 22, 2012.

2010 Mathematics Subject Classification. 20J05.

The second author was supported by the Research Fund K.U.Leuven and FWO-Flanders Research Fellowship.

INTERMEDIARIES IN BREDON (CO)HOMOLOGY AND CLASSIFYING SPACES 2

of Bredon cohomology of the stabilizer subgroups (see 3.5). As applications, we obtain several corollaries concerning the cohomological and geometric dimensions of E_{𝔉}G.

Secondly, using a construction similar to that of Kropholler’s class H𝔛 (see [13]), we introduce a new class of groups which is quite suited for applying this spectral sequence.

Definition 1.1. Let 𝔉 be a class of groups closed under taking subgroups and let G be a group. We set 𝔉 ∩ G = {H ≤ G | H is isomorphic to a subgroup in 𝔉}. Let 𝔛 be a class of groups. Then H^{𝔉}𝔛 is defined as the smallest class of groups containing the class 𝔛 with the property that if a group G acts cellularly on a finite dimensional CW-complex X with all isotropy subgroups in H^{𝔉}𝔛 and such that for each subgroup F ∈ 𝔉 ∩ G the fixed point set X^{F} is contractible, then G is in H^{𝔉}𝔛. LH^{𝔉}𝔛 is defined to be the class of groups that are locally H^{𝔉}𝔛-groups.

If 𝔉 contains the trivial group only, then LH^{𝔉}𝔛 is exactly Kropholler’s class LH𝔛. Perhaps surprisingly, when 𝔉 is the class of all finite groups, then LH^{𝔉}𝔉 also turns out to be quite large. It contains all elementary amenable groups and all linear groups over a field of characteristic zero. It is also closed under extensions, taking subgroups, amalgamated products, HNN-extensions, and countable directed unions. In view of this, we propose the following

Conjecture 1.2. Let 𝔉 be the class of finite or virtually cyclic groups. Then, H𝔉 = H^{𝔉}𝔉.

Next, we extend the notion of jump (co)homology to the context of Bredon (co)homology and prove Theorem (Theorem 6.3, Remark 6.5) An LH^{𝔉}𝔉-group G has jump 𝔉-Bredon cohomology if and only if G has a finite dimensional model for E_{𝔉}G, where 𝔉 is the class of finite groups or the class of virtually cyclic groups.

We note that the theorem does not hold if one removes the assumption that the group G is in LH^{𝔉}𝔉. Recently, Gandini showed that the first Grigorchuk group 𝔊 has jump cohomology of height one for the class of finite groups 𝔉 (see [7, Pr. of 4.7]). Yet he proved that 𝔊 cannot have a finite dimensional model for proper actions \underline{E}𝔊 since it has infinite rational cohomological dimension (see [7, 4.4]). Interestingly, this fact also allows one to conclude that 𝔊 is not in LH𝔉 (see [7, 4.8], [24, 3.2]). Based on this, we propose

Conjecture 1.3. Let G be a LH𝔉-group. Then G has jump 𝔉-Bredon cohomology if and only if G has a finite dimensional model for E_{𝔉}G, where 𝔉 is the class of finite groups or the class of virtually cyclic groups.

Lastly, we address the applications of our results to elementary amenable groups. In particular, we show

INTERMEDIARIES IN BREDON (CO)HOMOLOGY AND CLASSIFYING SPACES 3

Theorem (Theorem 7.4) If an elementary amenable group G acts freely and properly discontinuously on a manifold homotopy equivalent to a closed manifold, then G has a finite dimensional model for \underline{E}G.

2. Notations and Preliminaries

We begin with some notions and well-established facts concerning modules over the orbit category. To do homological algebra in the category of Bredon modules and in particular to define Bredon (co)homology, we follow the approach of Lück in [16], Chapter 9. Let us briefly recall some of the elementary notions we will need.

Let G be a group and let 𝔉 be a collection of subgroups of G closed under conjugation by every element of G and closed under taking subgroups. Then for every subgroup S of G, the collection 𝔉 ∩ S = {H ∩ S | H ∈ 𝔉} is contained in 𝔉. We refer to 𝔉 as a family of subgroups of G.

Let O_{𝔉}G denote the orbit category over 𝔉. A right O_{𝔉}G-module is a contravariant functor M : O_{𝔉}G → 𝔄𝔟. The category Mod_{𝔉}G is defined as follows: the objects are all the O_{𝔉}G-modules and the morphisms are all the natural transformations between the objects. Similarly, a left O_{𝔉}G-module is a covariant functor N : O_{𝔉}G → 𝔄𝔟 and G-Mod_{𝔉} is the category with objects the left O_{𝔉}G-modules and morphisms all the natural transformations between them.

For M ∈ Mod_{𝔉}G and N ∈ G-Mod_{𝔉}, we can define their tensor product over 𝔉 by:

M ⊗_{𝔉} N = ⨁_{H∈𝔉} M(G/H) ⊗ N(G/H)/ ∼

where we identify (m)M(φ) ⊗ n ∼ m ⊗ N(φ)(n) for each morphism (φ : G/H → G/K) ∈ O_{𝔉}G and elements m ∈ M(G/K), n ∈ N(G/H).

A sequence 0 → M′ → M → M′′ → 0 in Mod_{𝔉}G is called exact if it is exact after evaluating in G/H, for all H ∈ 𝔉. Now, let M ∈ Mod_{𝔉}G and consider the left exact functor:

Hom_{𝔉}(M, −) : Mod_{𝔉}G → 𝔄𝔟
N ↦ Hom_{𝔉}(M, N),

where Hom_{𝔉}(M, N) is the abelian group of all natural transformations from M to N. The O_{𝔉}G-module P is said to be projective if the functor Hom_{𝔉}(P, −) is exact. Just as for ℤG-modules, projective O_{𝔉}G-modules are flat, i.e. the functor P ⊗_{𝔉} − is exact.

The free Mod_{𝔉}G-module P_{K}^{G} = ℤ[map_{G}(−, G/K)] based at G/K ∈ O_{𝔉}G assigns to each G/H ∈ O_{𝔉}G the free ℤ-module generated by the set of all G-maps from G/H to G/K. In general, an O_{𝔉}G-module is said to be free if it is naturally isomorphic to a direct sum of free O_{𝔉}G-modules based at objects of O_{𝔉}G. An O_{𝔉}G-module P is projective if

INTERMEDIARIES IN BREDON (CO)HOMOLOGY AND CLASSIFYING SPACES 4

and only if it is a direct summand of a free module if and only if every exact sequence 0 → M → N → P → 0 splits (see [16, 9.20]).

It can be shown that Mod_{𝔉}G contains enough projective objects to construct projective resolutions. Hence, we can define n-th Bredon homology of G with coefficients in N ∈ G-Mod_{𝔉} to be:

H_{n}^{𝔉}(G, N) = Tor_{n}^{𝔉}(\underline{ℤ}, N) = H_{n}(P_{∗} ⊗_{𝔉} N),

where P_{∗} ↠ \underline{ℤ} is a projective O_{𝔉}G-resolution of the trivial O_{𝔉}G-module \underline{ℤ}. Also, n-th Bredon cohomology of G with coefficients in M ∈ Mod_{𝔉}G can be defined as:

H_{𝔉}^{n}(G, M) = Ext_{𝔉}^{n}(\underline{ℤ}, M) = H^{n}(Hom_{𝔉}(P_{∗}, M)).

Given M, N ∈ Mod_{𝔉}G, we denote by M ⊗ N the right O_{𝔉}G-module which assigns to each G/H ∈ O_{𝔉}G the ℤ-module M(G/H) ⊗ N(G/H) and to each morphism φ : G/H → G/K of O_{𝔉}G, the homomorphism:

M(φ) ⊗ N(φ) : M(G/K) ⊗ N(G/K) → M(G/H) ⊗ N(G/H).

For each subgroup S of G, let I : O_{𝔉∩S}S → O_{𝔉}G be the covariant inclusion functor:

S/H ⟼^{I^{ob}} G/H,
(S/H→^{x}S/K : H ↦ xK) ⟼^{I^{mor}} (G/H→^{x}H/K : H ↦ xK).

As for modules over group rings, associated to I, there are induction and restriction (additive) functors:

Res_{S}^{G} : Mod_{𝔉}G → Mod_{𝔉∩S}S
M ↦ M ∘ I,
Ind_{S}^{G} : Mod_{𝔉∩S}S → Mod_{𝔉}G
M ↦ M(?) ⊗_{𝔉∩S} ℤ[map_{G}(−, I(?))].

We will need the following relations between them.

Lemma 2.1 ([20], 3.5). Let 𝔉 be any given family of subgroups of G. Suppose S is a subgroup of G, the modules M, Q ∈ Mod_{𝔉}G and N ∈ G-Mod_{𝔉}. Then, there are natural isomorphisms:

(Q ⊗ Ind_{S}^{G}\underline{ℤ}) ⊗_{𝔉} N ≅ Res_{S}^{G}Q ⊗_{𝔉∩S} Res_{S}^{G}N,
Hom_{𝔉}(Q ⊗ Ind_{S}^{G}\underline{ℤ}, M) ≅ Hom_{𝔉∩S}(Res_{S}^{G}Q, Res_{S}^{G}M).

INTERMEDIARIES IN BREDON (CO)HOMOLOGY AND CLASSIFYING SPACES 5

3. Spectral sequence

In this section, we construct a spectral sequence associated to a G-CW-complex which is the Bredon analog of an E_{1}-term spectral sequence converging the (co)homology of G (see [4, p. 173-174]).

We begin with a basic lemma (see the proof of [20, 3.7]).

Lemma 3.1. Let 𝔉 be a family of subgroups of G. For any given subgroups F and H of G, there is an isomorphism:

Res_{F}^{G}Ind_{H}^{G} \underline{ℤ} ≅ ⨁_{x∈F \G/H} ℤ[map_{F} (−, F/H^{x^{−1}} ∩ F)].

In particular, if F ∈ 𝔉 or H ∈ 𝔉, then the module Res_{F}^{G}Ind_{H}^{G} \underline{ℤ} is free in Mod_{𝔉∩F} F.

Proof. This isomorphism readily follows from the fact that Ind_{H}^{G} \underline{ℤ} ≅ ℤ[map_{G}(−, G/H)] (see [20, 3.3]) and the isomorphism of F -sets, G/H ≅ ⨿_{x∈F \G/H}F/H^{x^{−1}} ∩ F. □

Proposition 3.2. Let 𝔉 be any given family of subgroups of G. Let R_{∗} ↠ \underline{ℤ} be a resolution in Mod_{𝔉}G such that, for each i, R_{i} ≅ ⨁_{σ∈Σ_{i}} Ind_{G_{σ}}^{G} \underline{ℤ}, where Σ_{i} is an index set and G_{σ} is a subgroup of G. Then, for modules N in G-Mod_{𝔉} and K, M in Mod_{𝔉}G, there are first quadrant spectral sequences:

E_{p,q}^{1}(N) = ⨁_{σ∈Σ_{p}}Tor_{q}^{𝔉∩G_{σ}}(Res_{G_{σ}}^{G}K, Res_{G_{σ}}^{G}N) ⟹ Tor_{p+q}^{𝔉}(K, N),

E_{1}^{p,q}(M) = ∏_{σ∈Σ_{p}}Ext_{𝔉∩G_{σ}}^{q}(Res_{G_{σ}}^{G}K, Res_{G_{σ}}^{G}M) ⟹ Ext_{𝔉}^{p+q}(K, M).

Proof. We derive the cohomological spectral sequence and note that the homological part is similar.

Let Q_{∗} ↠ K be a free resolution of the module K in Mod_{𝔉}G. By filtering the cochain complex:

C_{∗,∗∗} = Hom_{𝔉}(Q_{∗} ⊗ R_{∗∗}, M)

by rows and by columns, we respectively obtain the E_{2}- and E_{1}-term first quadrant spectral sequences:

IE_{2}^{p,q}(M) = H_{h}^{p}H_{v}^{q}(C_{∗,∗∗}) ⟹ H_{𝔉}^{p+q}(Tot(C_{∗,∗∗})),
IIE_{1}^{p,q}(M) = H_{h}^{q}(C_{∗,∗∗}) ⟹ H_{𝔉}^{p+q}(Tot(C_{∗,∗∗})).

We claim that, for all q > 0 and p ≥ 0,^{I}E_{2}^{p,q}(M) = 0 and^{I}E_{2}^{p,0}(M) = H^{p}(Hom_{𝔉}(Q_{∗}, M)) = Ext_{𝔉}^{p}(K, M). Therefore, H_{𝔉}^{p+q}(Tot(C_{∗,∗∗})) ≅ Ext_{𝔉}^{p+q}(K, M) for all p, q ≥ 0.

INTERMEDIARIES IN BREDON (CO)HOMOLOGY AND CLASSIFYING SPACES 6

To this end, suppose Q_{i} = ℤ[map_{G}(−, G/F)] for some G/F ∈ O_{𝔉}G and i ≥ 0. Then,

C_{i,∗∗} = Hom_{𝔉}(ℤ[map_{G}(−, G/F)] ⊗ R_{∗∗}, M)
≅ Hom_{𝔉}(Ind_{F}^{G}\underline{ℤ} ⊗ R_{∗∗}, M)
≅ Hom_{𝔉∩F} (Res_{F}^{G}R_{∗∗}, Res_{F}^{G}M)

By Lemma 3.1, it follows that Res_{F}^{G}R_{∗∗} is free in Mod_{𝔉∩F} F. Thus, Res_{F}^{G}R_{∗∗} ↠ \underline{ℤ} is a free resolution in Mod_{𝔉∩F} F. Hence, H^{q}(C_{i,∗∗}) = H_{𝔉∩F}^{q} (F, Res_{F}^{G}M) = 0 for all q > 0.

In general, since Q_{i} is a direct sum of free O_{𝔉}G-modules based at objects of O_{𝔉}G, this argument shows that H_{v}^{q}(C_{∗,∗∗}) = 0 for q > 0 and H_{v}^{0}(C_{∗,∗∗}) = Hom_{𝔉}(Q_{∗}, M) which finishes the claim.

Also, by Lemma 2.1, for each p ≥ 0 we have:

Hom_{𝔉}(Q_{∗} ⊗ R_{p}, M) ≅ ∏_{σ∈Σ_{p}} Hom_{𝔉}(Q_{∗} ⊗ Ind_{G_{σ}}^{G} \underline{ℤ}, M)

≅ ∏_{σ∈Σ_{p}} Hom_{𝔉∩G_{σ}}(Res_{G_{σ}}^{G}Q_{∗}, Res_{G_{σ}}^{G}M).

Since restriction functor respects projective modules (see [20, 3.7]), this entails:

IIE_{1}^{p,q}(M) = ∏_{σ∈Σ_{p}} H^{q}(Hom_{𝔉∩G_{σ}}(Res_{G_{σ}}^{G}Q_{∗}, Res_{G_{σ}}^{G}M))

= ∏_{σ∈Σ_{p}}Ext_{𝔉∩G_{σ}}^{q}(Res_{G_{σ}}^{G}K, Res_{G_{σ}}^{G}M).

By substituting the new terms into the E_{1}-term spectral sequence, we obtain the desired spectral sequence. □

Remark 3.3. When K is the trivial module \underline{ℤ}, we obtain the (co)homological spectral sequences:

E_{p,q}^{1}(N) = ⨁_{σ∈Σ_{p}}H_{q}^{𝔉∩G_{σ}}(G_{σ}, Res_{G_{σ}}^{G}N) ⟹ H_{p+q}^{𝔉}(G, N),

E_{1}^{p,q}(M) = ∏_{σ∈Σ_{p}}H_{𝔉∩G_{σ}}^{q}(G_{σ}, Res_{G_{σ}}^{G}M) ⟹ H_{𝔉}^{p+q}(G, M).

We will assume throughout that whenever a group G acts cellularly on a CW-complex, then the action of the stabilizer group of any cell fixes that cell pointwise.

Lemma 3.4. Let 𝔉 be any given family of subgroups of G. Suppose X is a G-CW-complex and denote by C_{∗∗}^{𝔉}(X, \underline{ℤ}) the Bredon chain complex of X. Then, for each i, we

INTERMEDIARIES IN BREDON (CO)HOMOLOGY AND CLASSIFYING SPACES 7

have:

C_{i}^{𝔉}(X, \underline{ℤ}) ≅ ⨁_{σ∈Σ_{i}} Ind_{G_{σ}}^{G} \underline{ℤ}.

where Σ_{i} is a set of representatives of all the G-orbits of i-cells of X and G_{σ} is the isotropy of σ.

Proof. For each H ∈ 𝔉, we obtain:

C_{i}^{𝔉}(X, \underline{ℤ})(G/H) = C_{i}(X^{H}, ℤ) ≅ ⨁_{σ∈Σ_{i}} ℤ[map_{G}(G/H, G/G_{σ})].

On the other hand, ℤ[map_{G}(−, G/G_{σ})] ≅ Ind_{G_{σ}}^{G} \underline{ℤ}, (see [20, 3.3]). Hence, the result is proved. □

Theorem 3.5. Let 𝔉 be any given family of subgroups of G. Suppose X is a G-CW-complex such that for each F ∈ 𝔉 the subcomplex X^{F} is contractible. Then, for modules N in G-Mod_{𝔉} and K, M in Mod_{𝔉}G, there are first quadrant spectral sequences:

E_{p,q}^{1}(N) = ⨁_{σ∈Σ_{p}}Tor_{q}^{𝔉∩G_{σ}}(Res_{G_{σ}}^{G}K, Res_{G_{σ}}^{G}N) ⟹ Tor_{p+q}^{𝔉}(K, N),

E_{1}^{p,q}(M) = ∏_{σ∈Σ_{p}}Ext_{𝔉∩G_{σ}}^{q}(Res_{G_{σ}}^{G}K, Res_{G_{σ}}^{G}M) ⟹ Ext_{𝔉}^{p+q}(K, M),

where Σ_{p} denotes a set of representatives of all the G-orbits of p-cells of X.

Proof. Let R_{∗} = C_{∗}^{𝔉}(X, \underline{ℤ}) be the Bredon chain complex associated to X. By our hypothesis it is exact and thus a resolution of \underline{ℤ}. The proof is now a direct consequence of Lemma 3.4 and Proposition 3.2. □

4. Applications

Here we present some immediate corollaries of the constructed spectral sequences. Although a number of these results have already been derived using several different techniques, we reprove them here to illustrate the versatility of our approach. Throughout this section, cd_{𝔉}(G) denotes the Bredon cohomological dimension of G with respect to the family of subgroups 𝔉. We substitute cd_{𝔉}(G) by either \underline{cd}(G) or \underline{\underline{cd}}(G) when 𝔉 is the family of finite or virtually cyclic subgroups, respectively.

As a first application of Theorem 3.5, we generalize the result obtained in [21, 3.4] to an arbitrary family 𝔉 and obtain an improved bound.

Corollary 4.1. Let 𝔉 be a family of subgroups of G. Suppose X is a G-CW-complex such that for each F ∈ 𝔉 the subcomplex X^{F} is contractible. Suppose also there is an integer b such that cd_{𝔉∩G_{v}}(G_{v}) ≤ b where v runs over all the vertices of X. Then,

cd_{𝔉}(G) ≤ b + dim(X).

Next, we present a new proof of a result of Martínez-Pérez (see [20, 4.1]) which can be seen as an algebraic analogue of a result of Luck and Weiermann (see [19, 5.1]).

Corollary 4.2. Let G be a group and 𝔉 ⊆ 𝔊 be families of subgroups of G. Suppose that there exists an integer d such that for every H ∈ 𝔊 we have cd_{𝔉∩H}(H) ≤ d. Then,

cd_{𝔉}(G) ≤ cd_{𝔊}(G) + d.

In particular, \underline{cd}(G) ≤ \underline{\underline{cd}}(G) + 1.

Proof. Let R_{∗} ↠ \underline{ℤ} be the restriction to Mod_{𝔉}G of a free resolution in Mod_{𝔊}G of length cd_{𝔊}(G). Note that R_{∗} ≅ ⨁_{σ∈Σ_{i}} Ind_{G_{σ}}^{G} \underline{ℤ}, where Σ_{∗} is an index set and each G_{σ} ∈ 𝔊. Now, an easy application of the cohomological spectral sequence of 3.3 for the family 𝔉 proves the first inequality.

From this and the fact that any virtually cyclic group has a 1-dimensional classifying space for proper actions we can deduce the second inequality. □

The following corollary has also been recently proven by Fluch and Nucinkis in [9, 3.1] using a different method (see also [8, 3.42]).

Corollary 4.3. Let G be a countable directed union of groups G_{i}. Let 𝔉 be a family of subgroups of G such that each H ∈ 𝔉 is a subgroup of some G_{j}, Then,

cd_{𝔉}(G) ≤ 1 + max{cd_{𝔉∩G_{i}}(G_{i})}.

Proof. Since G is countable, it follows by Bass-Serre theory [26] that there exists a G-tree X such that X^{G_{i}} ≠ ∅ for every G_{i}. Since each H ∈ 𝔉 is a subgroup of some G_{j}, one has that X^{G_{j}} ⊆ X^{H}. In particular, X^{H} is nonempty and hence it is connected, because if an element g ∈ G stabilizes two vertices of X then g stabilizes the (unique) path that connects them. This implies that X^{H} is contractible for every H ∈ 𝔉. Then, by Corollary 4.1, it follows that cd_{𝔉}(G) ≤ dim(X) + max{cd_{𝔉∩G_{i}}(G_{i})} as claimed. □

Remark 4.4. Note that the above conditions on the family of subgroups are satisfied when 𝔉 is the class of finite subgroups, virtually cyclic subgroups, or more generally consisting of groups that are subgroups of finitely generated subgroups of G.

Let \underline{gd}(G) denote the minimal dimension of all classifying spaces \underline{E}G of G. As a corollary, we deduce an algebraic analog of a theorem of Lück (see [17, 3.1]).

Corollary 4.5. Suppose 1 → H → G⟶^{π}Q → 1 is a short exact sequence of groups and suppose further there is an upper bound k on the order of the finite subgroups of Q. Then,

\underline{cd}(G) ≤ k(\underline{gd}(H)) + \underline{cd}(Q).

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Proof. For each finite subgroup F of Q, the subgroup π^{−1}(F) is an extension of H by F. By Serre’s construction (see the proof of Proposition 5.4 or [26]), it has a |F |(\underline{gd}(H))- dimensional model for \underline{E}π^{−1}(F). Hence, its Bredon cohomological dimension is bounded by k(\underline{gd}(H)). Now, applying the previous corollary ends the proof. □

Next, we derive a long exact sequence first obtained by Martínez-Pérez (see [20, 4.2]) which is the Bredon analogue of the long exact sequence in ordinary cohomology arising from a cellular action on a tree.

Corollary 4.6. Suppose 𝔉 is a family of subgroups of G and X is a G-tree such that for each F ∈ 𝔉, X^{F} is nonempty. Denote by G_{v} the stabilizer of a vertex v and by G_{e} the stabilizer of an edge e. Let Σ_{0} and Σ_{1} be the sets of G-orbit representatives of the vertexes and the edges of X, respectively. Then, for every M in Mod_{𝔉}G, there is a long exact sequence:

⋯ → H_{𝔉}^{i}(G, M) → ⨁_{v∈Σ_{0}}H_{𝔉∩G_{v}}^{i}(G_{v}, Res_{G_{v}}^{G}M) → ⨁_{e∈Σ_{1}}H_{𝔉∩G_{e}}^{i}(G_{e}, Res_{G_{e}}^{G}M) → H_{𝔉}^{i+1}(G, M) → ⋯

Proof. Let us consider the spectral sequence of 3.5 applied to the G-CW-complex X and K = \underline{ℤ}. On the first page there are only two possibly nonzero vertical lines, i.e. E_{1}^{p,q}(M) = 0 for p > 1. Therefore, we have a Wang sequence (see for example [28, p.130]):

⋯ → H_{𝔉}^{i}(G, M) → E_{1}^{0,i}(M) → E_{1}^{1,i}(M) → H_{𝔉}^{i+1}(G, M) → ⋯

Substituting the E_{1}-terms, yields the desired long exact sequence. □

Consequently, we obtain a Mayer-Vietoris sequence for amalgamated products.

Corollary 4.7. Let G = A∗_{C} B, where A, B and C are groups and suppose M in Mod_{𝔉}G. Suppose, every F in 𝔉 is a subgroup of a conjugate of A or B. Then, there is a long exact sequence:

⋯ → H_{𝔉}^{i}(G, M) → H_{𝔉∩A}^{i}(A, Res_{A}^{G}M)⊕H_{𝔉∩B}^{i}(B, Res_{B}^{G}M) → H_{𝔉∩C}^{i}(C, Res_{C}^{G}M) → H_{𝔉}^{i+1}(G, M) ⋯

Proof. Let X be the Bass-Serre tree of G. The result immediately follows from Corollary 4.6. □

Remark 4.8. This Mayer-Vietoris sequence has already been derived in [22, 3.32] by a different approach, using a homotopy-push-out of the induced spaces of the classifying spaces corresponding to the subgroups in the amalgamated product.

5. H^{𝔉}𝔛-groups

We begin with the definition of a class of groups which is analogous to Kropholler’s definition of the class H𝔛 (see [13]).

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Unless otherwise specified, 𝔉 will denote a subgroup closed class of groups. Note that for any group G the collection:

𝔉 ∩ G = {H ≤ G | H is isomorphic to a subgroup in 𝔉}

is then a family of subgroups of G. If 𝔉 is the class of finite groups, when referring to Bredon (co)homology of G with respect to 𝔉∩G, we will simply say Bredon (co)homology of G.

Definition 5.1. Let 𝔛 be a class of groups. H^{𝔉}𝔛 is defined as the smallest class of groups containing the class 𝔛 with the property that if a group G acts cellularly on a finite dimensional CW-complex X with all isotropy subgroups in H^{𝔉}𝔛 and such that for each subgroup F ∈ 𝔉 ∩ G the fixed point set X^{F} is contractible, then G is in H^{𝔉}𝔛.

Remark 5.2. When 𝔉 is the trivial class, then H^{𝔉}𝔛 is exactly the Kropholler’s class H𝔛. Also, note that X need not be a model for E_{𝔉}G, because we do not require for X^{F} to be empty if F is not in 𝔉. It is, in a way, an intermediary complex.

There is also an inductive definition of H^{𝔉}𝔛-groups, by using ordinal numbers:

(a) Let H_{0}^{𝔉}𝔛 = 𝔛.

(b) For ordinal β > 0, define H_{β}^{𝔉}𝔛 to be the class of groups that can act cellularly on a finite dimensional CW-complex X such that each isotropy group is in H_{α}^{𝔉}𝔛 for some α < β (α can depend on the isotropy group) and for each subgroup F ∈ 𝔉 ∩ G the fixed point set X^{F} is contractible. We denote by LH^{𝔉}𝔛 the class of locally H^{𝔉}𝔛-groups, i.e. groups whose finitely generated

subgroups belong to H^{𝔉}𝔛.

Remark 5.3. It immediately follows:

- a group is in H^{𝔉}𝔛 if and only if it belongs to H_{α}^{𝔉}𝔛 for some ordinal α;

- H_{α}^{𝔉}𝔛 ⊆ H_{α}𝔛 for each ordinal α where H_{α}𝔛 is the Kropholler’s class for the ordinal α;

- G admits a finite dimensional model for E_{𝔉}G if and only if it is in H_{1}^{𝔉}𝔉.

Proposition 5.4. H^{𝔉}𝔛 is subgroup closed. If 𝔉 is the class of finite groups, then H^{𝔉}𝔉 is extension closed.

Proof. It is obvious that H_{α}^{𝔉}𝔛 is subgroup closed for every ordinal α, so H^{𝔉}𝔛 is subgroup closed.

The proof that H^{𝔉}𝔉 is extension closed is analogous to the proof of 2.3 in [13]. First we prove that H_{α}^{𝔉}𝔉 is closed under finite extensions, using induction on the ordinal α. This is obvious for α = 0. Assume that it is true for all β < α and let G be a group with a finite index subgroup N that belongs to H_{α}^{𝔉}𝔉. There is a finite dimensional N-CW-complex X such that each isotropy group is in H_{β}^{𝔉}𝔉 for some β < α and X^{F} is contractible for any finite subgroup F of N. Serre’s construction (cf. [4]) shows that the G-set Y = Hom_{G}(N, X) has the structure of a CW-complex via the bijection φ : Y → ∏_{t∈T} X, with φ(f) = (f(t))_{t∈T} where T is a set of coset representatives of N in G. If K is the isotropy group of an element f of Y then f(g) = f(gk) for all k ∈ K and g ∈ G. In particular f(e) = f(k) = kf(e) for all k ∈ N ∩ K, so N ∩ K is a subgroup of the isotropy group of f(e), which belongs to H_{β}^{𝔉}𝔉 for some β < α. Since H_{β}^{𝔉}𝔉 is subgroup closed it follows from the inductive hypothesis that K belongs to H_{β}^{𝔉}𝔉. Next we show that Y^{F} is contractible for every finite subgroup F of G. For this, take a set S of double coset representatives, such that G = ∪_{s∈S}NsF and let F_{s} = N ∩ sF s^{−1}. The map ψ : Y^{F} → ∏_{s∈S} X^{F_{s}} with ψ(f) = (f(s))_{s∈S} is well defined because if f ∈ Y^{F} and g ∈ F_{s} we have s^{−1}gs ∈ F, so gf(s) = f(gs) = f(ss^{−1}gs) = (s^{−1}gsf)(s) = f(s), thus f(s) ∈ X^{F_{s}}. Obviously ψ is a bijection and each X^{F_{s}} is contractible, so Y^{F} is contractible.

Now, we prove by induction on the ordinal β that if N is an H_{α}^{𝔉}𝔉-subgroup of G such that G/N is in H_{β}^{𝔉}𝔉 then G is in H^{𝔉}𝔉. Indeed, the previous argument proves this claim for β = 0. Let β > 0 and assume that the claim holds for γ < β. Let X be a G/N-CW-complex with stabilizers in H_{γ}^{𝔉}𝔉 for γ < β and X^{F} be contractible for any finite subgroup F of G/N. Then G acts on X via the projection π : G → G/N with cell stabilizers that are extensions of N by H_{γ}^{𝔉}𝔉-groups for γ < β. It follows from the inductive hypothesis that these belong to H^{𝔉}𝔉. Now, if F is a finite subgroup of G then X^{F} = X^{π(F)} is contractible since π(F) is a finite subgroup of G/N. Thus, G belongs to H^{𝔉}𝔉. □

Proposition 5.5. If 𝔉 is the class of finite groups, then H^{𝔉}𝔉 is closed under countable directed unions, amalgamated products and HNN-extensions.

Proof. Let G = ⋃_{n∈ℕ} G_{n} where G_{n} ≤ G_{n+1} and G_{n} ∈ H^{𝔉}𝔉 for all n ∈ ℕ. Then it follows from Bass-Serre theory that G acts on a tree T such that T^{G_{n}} ≠ ∅ for every G_{n}. If F is a finite subgroup of G then F ≤ G_{n} for some n ∈ ℕ, so T^{F} ≠ ∅. Thus T^{F} is contractible.

If G is an amalgamated product A ∗_{C} B or an HNN-extension K∗_{α}, then it follows again from Bass-Serre theory that G acts on a tree T such that T^{A} ≠ ∅, T^{B} ≠ ∅ and T^{K} ≠ ∅, respectively. The same is true for every conjugate of A, B and K. Now, any finite subgroup F of G is conjugate to a subgroup of A or B, and K respectively, so T^{F} ≠ ∅ and hence it is contractible. □

Theorem 5.6. If 𝔉 is the class of finite groups, then H^{𝔉}𝔉 contains all countable elementary amenable groups and all countable linear groups over a field of characteristic 0.

Proof. Elementary amenable groups: H^{𝔉}𝔉 contains all finitely generated abelian groups. Now, recalling Kropholler, Linnel and Moody’s inductive definition of elementary amenable

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groups (see [14]), by transfinite induction, using extensions and countable directed union closure, it follows that H^{𝔉}𝔉 contains all countable elementary amenable groups.

Linear groups: Since H^{𝔉}𝔉 is closed under countable directed unions, it suffices to consider finitely generated linear groups. In [13, 2.5], Kropholler proves that countable linear groups over a field of characteristic 0 belong to H𝔉, making crucial use of results of Alperin and Shalen (see [1]). Since torsion free H𝔉-groups belong to H^{𝔉}𝔉, it follows that torsion free linear groups belong to H^{𝔉}𝔉. Now, by a theorem of Borel (cf. Theorem 1.5 in [1]) every finitely generated linear group is virtually torsion free. Since H^{𝔉}𝔉 is closed under finite extensions, the result follows. □

6. Finiteness Properties

A weaker notion than (co)homological dimension called (co)homological jump height has been introduced in [24] and many of its finiteness properties have been considered in [25]. When studying group actions on certain complexes, we have seen that it is often more natural to utilize jump height rather than (co)homological dimension. Here, we extend the definition of jump (co)homology and jump height to Bredon cohomology and consider some applications.

Definition 6.1. A discrete group G has jump 𝔉-Bredon (co)homology if there exists an integer k ≥ 0, such that for each subgroup H of G we have hd_{𝔉∩H}(H) = ∞ (cd_{𝔉∩H}(H) = ∞) or hd_{𝔉∩H}(H) ≤ k (cd_{𝔉∩H}(H) ≤ k). The minimum of all such k is called jump height and denoted hjh_{𝔉}(G) (cjh_{𝔉}(G)).

First, we need a generalization of Benson’s lemma (see [5, 5.6]) that applies in our context.

Lemma 6.2. Suppose that G_{α} (0 ≤ α < γ, G_{0} = {e}) is an ascending chain of groups with union G = ∪_{α<γ}G_{α}, for some ordinal γ. Let 𝔉 be a family of subgroups of G such that every H ∈ 𝔉 is a subgroup of some G_{α}. If M is a module in Mod_{𝔉}G whose restriction to each subgroup G_{α} is projective in Mod_{𝔉∩G_{α}}G_{α}, then M has projective dimension at most one.

Proof. Denote Z_{G_{α}} = ⨁_{δ≥α} ℤ[map_{G}(−, G/G_{δ})]. Since M ⊗ Z_{G_{α}} ≅ ⨁_{δ≥α} Ind_{G_{δ}}^{G}Res_{G_{δ}}^{G}M, by adjointness of induction and restriction and definition of projectives, we can deduce that Q_{α} := M ⊗ Z_{G_{α}} is projective.

Given α ≤ β, we define an epimorphism ψ_{α}^{β} : Z_{G_{α}} → Z_{G_{β}} as follows: the summand ℤ[map_{G}(−, G/G_{δ})] of Z_{G_{α}} is mapped to the summand ℤ[map_{G}(−, G/G_{μ})] of Z_{G_{β}}, where μ = max{β, δ}, by the natural quotient map G/G_{δ} → G/G_{μ}. Note that ψ_{α}^{β} induces an epimorphism id_{M} ⊗ ψ_{α}^{β} : Q_{α} → Q_{β}.

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Let P_{α} be the kernel of id_{M} ⊗ ψ_{0}^{α} : Q_{0} → Q_{α}. If α ≤ β < γ, then P_{α} ≤ P_{β}. Since Q_{β} is projective, the epimorphism id_{M} ⊗ ψ_{α}^{β} : Q_{α} → Q_{β} must split. Then, the kernel which is isomorphic to P_{β}/P_{α}, is projective.

Also, it follows from our hypothesis that for each α, there exists an epimorphism from Z_{G_{α}} onto ℤ[map_{G}(−, G/G)] ≅ \underline{ℤ} that maps each summand ℤ[map_{G}(−, G/G_{δ})] of Z_{G_{α}} to ℤ[map_{G}(−, G/G)] via the trivial quotient map G/G_{δ} → G/G. This epimorphism induces an epimorphism ϕ_{α} : Q_{α} → M. Note that for each α ≤ β, we have ϕ_{β} ∘ (id_{M} ⊗ ψ_{α}^{β}) = ϕ_{α}.

Let P_{γ} the kernel of the induced epimorphism from Q_{0} to M. The fact that \underrightarrow{lim} Z_{G_{α}} = \underline{ℤ}, implies that \underrightarrow{lim} Q_{α} = M. So, we can deduce that P_{γ} = ∪_{α<γ}P_{α}. Now, by using transfinite induction, it is not difficult to see that,

P_{γ} ≅ ⨁_{α<γ} P_{α+1}/P_{α}

and hence it is projective. Since M ≅ Q_{0}/P_{γ}, it has projective dimension at most one. □

Following the notation of the previous section, let hd_{𝔉∩G}(G) (cd_{𝔉∩G}(G)) denote the Bredon (co)homological dimension of the group G with respect to a class of groups 𝔉.

The next result is a generalization of Theorem 3.3 of [24], Theorem 8 of [27], and of Proposition 3.1 of [12].

Theorem 6.3. Let 𝔉 be a subgroup closed class of groups and let 𝔛 be a class of groups with finite Bredon (co)homological dimension with respect to 𝔉. Suppose G is in H^{𝔉}𝔛 or 𝔉 ∩ G consists of groups that are subgroups of finitely generated subgroups of G and G is in LH^{𝔉}𝔛. Then G has finite Bredon (co)homological dimension with respect to 𝔉 if and only if G has jump Bredon (co)homology with respect to 𝔉.

Proof. We will prove the result for cohomology. The homological version is analogous.

One direction is trivial. So, let G have jump Bredon cohomology with respect to 𝔉. Suppose G is in H^{𝔉}𝔛. Since G is in H_{α}^{𝔉}𝔛 for some ordinal α, we can assume by transfinite induction, there exists a finite dimensional G-CW-complex X such that for each F ∈ 𝔉 ∩ G the subcomplex X^{F} is contractible and that each isotropy group has finite Bredon cohomological dimension bounded by cjh_{𝔉}(G). Now, applying Theorem 3.5, we can deduce that H_{𝔉}^{i}(G, M) = 0 for all i > cjh_{𝔉}(G) + dim(X). Therefore, G has finite Bredon cohomological dimension equal to cjh_{𝔉}(G).

Suppose G is in LH^{𝔉}𝔛 and 𝔉 ∩ G consists of groups that are subgroups of finitely generated subgroups of G. Let P_{∗} ↠ \underline{ℤ} be a projective resolution of \underline{ℤ} in Mod_{𝔉}G and denote d = cjh_{𝔉}(G). We first show that the module P_{d} has projective dimension at most

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If G is countable, then there is an ascending chain of finitely generated subgroups G_{i}, i ∈ ℕ, such that G = ∪_{i=1}^{∞}G_{i}. Since, each G_{i} belongs to the class H^{𝔉}𝔛, we have cd_{𝔉∩G_{i}}(G_{i}) ≤ d. So, Lemma 6.2 implies that P_{d} has projective dimension at most 1.

If G is uncountable, we proceed by induction on the cardinality of G. The group G can be expressed as an ascending union of subgroups, G = ∪_{α<γ}G_{α} for some ordinal γ, such that each subgroup G_{α} has strictly smaller cardinality than G. By induction, cd_{𝔉∩G_{α}}(G_{α}) ≤ d for each α. Again, applying Lemma 6.2, shows that the module P_{d} has projective dimension at most 1.

Since P_{d} has projective dimension at most 1, the trivial module \underline{ℤ} has projective dimension at most d + 1. It follows that cd_{𝔉}(G) ≤ d + 1, and hence cd_{𝔉}(G) = d. □

Remark 6.4. We give an alternative proof of the first part of this result.

Let X be a G-CW-complex such that for each F ∈ 𝔉 ∩ G the subcomplex X^{F} is contractible and that each isotropy group has finite 𝔉-Bredon cohomological dimension bounded by cjh_{𝔉}(G). Since X^{F} is contractible for each F ∈ 𝔉 ∩ G, the chain complex C_{∗}^{𝔉}(X, \underline{ℤ}) is exact. By the induction, pl_{𝔉∩G_{σ}}(\underline{ℤ}) ≤ cjh_{𝔉}(G) for each G_{σ}. Since C_{i}^{𝔉}(X, \underline{ℤ}) ≅ ⨁_{σ∈Σ_{i}} Ind_{G_{σ}}^{G} \underline{ℤ}, we obtain pl_{𝔉∩G_{σ}}(C_{i}^{𝔉}(X, \underline{ℤ})) ≤ cjh_{𝔉}(G) for each i. It follows that pl_{𝔉}(\underline{ℤ}) ≤ cjh_{𝔉}(G) + dim(X). Since G has jump Bredon cohomology, this shows pl_{𝔉}(\underline{ℤ}) = cjh_{𝔉}(G).

Remark 6.5. Note that by a result of Lück [16], Theorem 6.3 establishes in particular that:

(i) every H^{𝔉}𝔉-group has jump 𝔉-Bredon cohomology if and only if G has a finite dimensional model for E_{𝔉}G.

(ii) every LH^{𝔉}𝔉-group has jump 𝔉-Bredon cohomology if and only if G has a finite dimensional model for E_{𝔉}G, where 𝔉 is the class of finite groups or the class of virtually cyclic groups.

It follows from Theorem 5.6 that when 𝔉 is the class of finite groups, the class LH^{𝔉}𝔉 contains all elementary amenable groups and all linear groups of characteristic zero. Applying Theorem 6.3, we obtain the following corollary which is a generalization of a result of Jo and Nucinkis (see [12, 3.2])

Corollary 6.6. Let G be an elementary amenable group or a linear group of characteristic zero. Then G has jump 𝔉-Bredon cohomology if and only if G has a finite dimensional model for E_{𝔉}G, where 𝔉 is the class of finite groups.

Definition 6.7. A group G has periodic 𝔉-Bredon cohomology with period q > 0 after k steps, if the functors H_{𝔉}^{n}(G, −) and H_{𝔉}^{n+q}(G, −) are naturally equivalent for all n > k.

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Using the Eckmann-Shapiro lemma it is easily proved that periodic Bredon cohomology after some steps is a subgroup closed property, i.e. every subgroup S of G has periodic 𝔉 ∩ S-Bredon cohomology with period q after k steps.

Corollary 6.8. Let G be in LH^{𝔉}𝔉, where 𝔉 is the class of finite groups. Then G has periodic Bredon cohomology after some steps if and only if G admits a finite dimensional model for \underline{E}G.

Proof. Assume G has periodic Bredon cohomology with period q after k steps, and let S be a subgroup with \underline{cd}(S) = r. Then H_{𝔉∩S}^{r}(S, M) ≠ 0 for some O_{𝔉∩S}S-module M. If r > k then H_{𝔉∩S}^{r+q}(S, M) ≅ H_{𝔉∩S}^{r}(S, M) ≠ 0 a contradiction, since H_{𝔉∩S}^{r+q}(S, −) = 0. It follows that r ≤ k, so G has jump Bredon cohomology, and applying Theorem 6.3 and Lück’s theorem [16] we have that G admits a finite dimensional model for \underline{E}G.

The converse is trivial, since H_{𝔉}^{n}(G, −) = 0 = H_{𝔉}^{n+q}(G, −) for any q > 0 and n > \underline{cd}(G). □

7. Applications for elementary amenable groups

Lastly, we derive some corollaries of Theorem 6.3 regarding elementary amenable groups.

Proposition 7.1. Let 𝔽 be a field of characteristic zero and suppose G is an elementary amenable group. Then, the conditions that G has finite Hirsch length, G has finite homological dimension over 𝔽, G has jump homology over 𝔽, G has jump Bredon homology, and G has finite Bredon homological dimension are equivalent. Moreover,

h(G) = hd_{𝔽}(G) = hjh_{𝔽}(G) = \underline{hd}(G) = \underline{hjh}(G).

Proof. By the analogues of Proposition 4.3 of [23] and Lemma 2 of [11] (see [10, p.5]), \underline{hd}(G) ≥ hd_{𝔽}(G) ≥ h(G). But, by Theorem 1 of Flores and Nucinkis in [10], h(G) = \underline{hd}(G).

The second equality is a consequence of Theorem 6.3 assuming 𝔉 is the trivial class, 𝔛 is the class of finite groups and the homology is considered over the field 𝔽. The last equality follows again from Theorem 6.3 where 𝔛 = 𝔉 is the class of finite groups. □

The next theorem is a generalization of Corollary 4.3 of [12].

Proposition 7.2. Suppose 𝔽 be a field of characteristic zero and G is an elementary amenable group. Let X be an n-dimensional G-CW-complex. Suppose there exists an integer k such that for each i > k, H_{i}(X, 𝔽) = 0 and H_{k}(X, ℤ) ≅ ℤ^{m} ⊕ A where m > 0 and A is a finite group. If all the stabilizer subgroups of the action of G on X have finite Bredon homological dimension uniformly bounded by an integer b, then

\underline{hd}(G) ≤ b + n − k + \frac{1}{2}m(m − 1).

Proof. If G is elementary amenable, by Proposition 5.1 of [25] and Proposition 7.1,

hjh_{𝔽}(G) < b + n − k + \frac{1}{2}m(m − 1). Again applying 7.1, finishes the proof. □

Remark 7.3. Note that the assumptions of Theorem 7.2 can occur quite naturally; for instance, when X is a proper G-CW-complex which is finitely dominated or homotopy equivalent to a closed manifold. In particular, we can deduce the following.

Theorem 7.4. Suppose G is an elementary amenable group acting freely and properly discontinuously on an n-dimensional manifold M homotopy equivalent to a closed manifold N of dimension k. Then \underline{cd}(G) ≤ n − k + 1.

Proof. We first argue that we can, without loss of generality, assume that the closed manifold N is orientable. Indeed, suppose it is not and consider the oriented double covering \overline{N} of N. Then, its pullback by the homotopy equivalence from M to N gives us a double covering \overline{M} of M and a homotopy equivalence from \overline{M} to \overline{N}. Since G acts freely on M, we can deduce that there exists a group \overline{G} acting freely on \overline{M} such that ℤ_{2} ⊴ \overline{G} and \overline{G}/ℤ_{2} ≅ G. But, in this case, \underline{cd}(G) = \underline{cd}(\overline{G}) (see [23, 5.5]). So, we can assume that N is orientable.

Now, since H_{k}(M, ℤ) = ℤ and H_{i}(M, 𝔽) = 0 for i > k, where 𝔽 is a field of characteristic 0, we have that \underline{hd}(G) ≤ n − k. Hence, \underline{cd}(G) ≤ n − k + 1, because G is countable (see [23, 4.1]). □

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Department of Mathematics, University of Athens, Athens, Greece

E-mail address: fdebeg@math.uoa.gr

Department of Mathematics, Katholieke Universiteit Leuven, Kortrijk, Belgium

E-mail address: Nansen.Petrosyan@kuleuven-kortrijk.be

Department of Mathematics, University of Athens, Athens, Greece

E-mail address: otalelli@math.uoa.gr